Problem 46 - Thomas Calculus

Question

Exercises \(45-48\) give equations for parabolas and tell how many units up or down and to the right or left each parabola is to be shifted. Find an equation for the new parabola, and find the new vertex, focus, and directrix. $$ y^{2}=-12 x, \quad \text { right } 4, \text { up } 3 $$

Step-by-Step Solution

Answer

New equation: \((y-3)^2 = -12(x-4)\); vertex: (4, 3); focus: (1, 3); directrix: \(x=7\).

Step 1: Identify Original Parabola Components

The given equation is \( y^2 = -12x \). This is the standard form of a leftward-opening parabola, \( y^2 = -4px \), where \( 4p = 12 \). Thus, \( p = 3 \) and the vertex is at the origin \((0,0)\). The focus is at \((-p, 0) = (-3, 0)\) and the directrix is at \(x = p = 3\).

Step 2: Calculate New Vertex

Shift the vertex of the parabola right by 4 units and up by 3 units. The original vertex is \((0, 0)\). After shifting, the new vertex is \((0+4, 0+3) = (4, 3)\).

Step 3: Formulate New Parabola Equation

To find the new equation in vertex form, substitute the new vertex, \((h, k) = (4, 3)\), into the vertex form of a parabola, \((y-k)^2 = -4p(x-h)\). We know \(p = 3\), so the new equation is \((y-3)^2 = -12(x-4)\).

Step 4: Determine New Focus and Directrix

The focus of the original parabola was at \((-3, 0)\), and shifting it right by 4 and up by 3 results in \((-3+4, 0+3) = (1, 3)\). The directrix was originally \(x = 3\), and moving it right by 4 units results in the new directrix \(x = 3+4 = 7\).

Key Concepts

Vertex Form Equation

When working with parabolas, the vertex form of the equation is immensely helpful in identifying the position and shape of the parabola quickly. The vertex form equation is given by

• \((y - k)^2 = 4p(x - h)\)

where

• \((h, k)\) represents the vertex of the parabola,
• \(p\) is the distance from the vertex to the focus, and it's the same distance to the directrix,
• \(x\) and \(y\) are the variables in the coordinate plane.

In this form, the parabola can open to the left, right, up, or down, depending on which variable is squared and the sign of the coefficient. For example, a negative sign before the quadratic term indicates a left or downward opening. In our given example, the transformed vertex form after the shifts is

• \((y-3)^2 = -12(x-4)\)

This tells us the parabola opens to the left, with its vertex now located at \((4, 3)\). By understanding the vertex form, you can easily sketch the parabola and determine its general direction and position.

Focus of a Parabola

The focus is a significant point in the structure of a parabola. It's one of the parameters that determine the steepness and direction of the parabola. In the context of the vertex form, calculating the focus involves using the distance \(p\) as follows:

• Add the distance \(p\) to the x-coordinate (or y-coordinate, if vertical) of the vertex.
• Use the new value to calculate the exact position of the focus.

For the given problem, the original focus was at \((-3, 0)\) since the parabola opens leftward. After the vertex was shifted to \((4, 3)\), the focus followed the same transformation rules:

• Shift right: a change in x-coordinate by 4 units,
• Shift up: a change in y-coordinate by 3 units,

Thus, the new focus position is

• \((1, 3)\).

Knowing the location of the focus helps in the accurate drawing and analysis of the parabola. It also aids in understanding how light or signals reflect off the parabola in practical real-world applications like satellite dishes.

Directrix of a Parabola

The directrix is a crucial line associated with a parabola, located at a distance \(p\) from the vertex, opposite to the focus. It runs perpendicular to the axis of symmetry of the parabola. For a parabola that opens leftward or rightward, the directrix is a vertical line, while for upward or downward-opening parabolas, the directrix is horizontal.
In the problem we solved, since the original parabola's directrix was at \(x = 3\) and opened leftward, the transformation rules apply similarly to the directrix:

• Since the parabola is moved 4 units to the right, the directrix also shifts right by 4 units.

Hence, the new position of the directrix becomes

• \(x = 7\).

The directrix serves as a reference line from which every point on the parabola is equidistant along with the focus, ensuring the perfect shape of the parabola. Understanding the directrix aids in comprehensive studies involving reflective properties and geometrical applications in mathematics and physics.